Vector Algebra - Practice-2.1 - Numbas at mathcentre.ac.uk

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Created this as a copy of Vector Algebra - Practice-2.

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Question statement

Let $\var{vi},\var{vj},\var{vk}$ form an orthonormal basis. Consider

$\var{va}=\var{disp(a)}, \qquad \var{vb}= \var{disp(b)} , \qquad \var{vc}=\var{{ccoef}*a[0]}\var{vi} -\var{lcoef} t \var{vj}-\var{abs({ccoef}*a[2])}\var{vk},$
where

$t\in\mathbb{R}$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

va	string	\simplify\{ underline:boldsymb
vb	string	\simplify\{ underline:boldsymb
vc	string	\simplify\{ underline:boldsymb
vd	string	\simplify\{ underline:boldsymb
ve	string	\simplify\{ underline:boldsymb
vi	string	\simplify\{ underline:boldsymb
vj	string	\simplify\{ underline:boldsymb
vk	string	\simplify\{ underline:boldsymb
vn	string	\simplify\{ underline:boldsymb
vr	string	\simplify\{ underline:boldsymb
vv	string	\simplify\{ underline:boldsymb
vah	string	\simplify\{ hat:underline:bold

			Name	Type	Generated Value

a	vector	vector(-2,1,-2)
acoef	integer	1
alen	number	3
apar	vector	vector(2,-1,2)
aset	list	[ 2, 1, 2 ]
asign	list	[ -1, 1, -1 ]
b	vector	vector(1,-4,-8)
blen	number	9
bset	list	[ 1, 4, 8 ]
bsign	list	[ 1, -1, -1 ]
ccoef	integer	2
costheta	list	[ 10, 27 ]
l	list	[ -2, 3 ]
lcoef	integer	3
lort	list	[ -7, 3 ]
mark	integer	1
dop	number	-36
dop1	number	-12

			Name	Type	Generated Value

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
latex('\\simplify\{ underline:boldsymbol:a \}')

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

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Generated value: `string`

\simplify\{ underline:boldsymbol:a \}

This variable doesn't seem to be used anywhere.

Parts

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2. Gap Gap 1. Number entry
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Part b) Gap-fill
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Part c) Gap-fill
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Part d) Gap-fill
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Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

Find the vector $\var{vd}$ whose magnitude is $\simplify{{acoef}*{alen}}$ and which is opposite to $\var{va}$ :

$\var{vd}=$ Gap 0. $\var{vi}+$ Gap 1. $\var{vj}+$ Gap 2. $\var{vk}$ .

Advice

a) Let $\var{vah}$ be the unit vector corresponding to $\var{va}$ , i.e.

$\var{vah}=\frac{1}{|\var{va}|}\var{va}.$
Since

$\var{vd}$ is opposite to

$\var{va}$ , it is also opposite to

$\var{vah}$ . Since

$|\var{vd}|=\simplify{{acoef}*{alen}}$ , we conclude that

$\var{vd} = -\simplify{{acoef}*{alen}}\var{vah}=-\frac{\simplify{{acoef}*{alen}}}{|\var{va}|}\var{va}.$
We have:

$|\var{va}|=\sqrt{\var{abs(a[0])}^2+\var{abs(a[1])}^2+\var{abs(a[2])}^2}=\var{alen}.$
Therefore,

$\var{vd} =-\frac{\simplify{{acoef}*{alen}}}{\var{alen}}\bigl(\var{disp(a)}\bigr)=\var{disp(apar)}.$

b) We know that

$\cos \theta =\frac{\var{va}\cdot\var{vb}}{|\var{va}|\, |\var{vb}|}.$
We have found that

$|\var{va}|=\var{alen}$ . Next,

$|\var{vb}|=\sqrt{\var{abs(b[0])}^2+\var{abs(b[1])}^2+\var{abs(b[2])}^2}=\var{blen}.$
We have also that

$\var{va}\cdot\var{vb}=\bigl(\var{disp(a)}\bigr)\cdot\bigl(\var{disp(b)}\bigr)=\var{dot(a,b)}$
Therefore,

$\cos \theta=\frac{\var{dot(a,b)}}{\var{alen} \cdot\var{blen}}=\var{sfrac(costheta[0],costheta[1])}.$

c) Since vectors $\var{va}=\var{disp(a)}$ and $\var{vc}=\var{{ccoef}*a[0]}\var{vi} -\var{lcoef} t \var{vj}+\var{abs({ccoef}*a[2])}\var{vk}$ are parallel, there exists a number $r\in\mathrm{R}$ such that

$\var{vc}=r\var{va}.$
Equating coefficients before

$\var{vi},\var{vj},\var{vk}$ , we get

$\begin{cases} \displaystyle \var{{ccoef}*a[0]} = \simplify{{a[0]}*r},\\[2mm] \displaystyle -\var{lcoef} t =\simplify{{a[1]}*r},\\[2mm] \displaystyle \var{abs({ccoef}*a[2])} =\simplify{{a[2]}*r}, \end{cases} \qquad\qquad \begin{cases} \displaystyle r=\var{ccoef},\\[3mm] \displaystyle t = -\frac{ \simplify{{a[1]}*r}}{\var{lcoef}}, \end{cases} \qquad\qquad t=\var{sfrac(l[0],l[1])}.$

d) Vectors $\var{vb}= \var{disp(b)}$ and $\var{vc}=\var{{ccoef}*a[0]}\var{vi} -\var{lcoef} t \var{vj}+\var{abs({ccoef}*a[2])}\var{vk}$ are orthogonal if and only if

$\var{vb}\cdot\var{vc}=0.$
Since

$\var{vb}\cdot\var{vc}=\bigl( \var{disp(b)}\bigr) \cdot\bigl( \var{{ccoef}*a[0]}\var{vi} -\var{lcoef} t \var{vj}+\var{abs({ccoef}*a[2])}\var{vk}\bigr) =\simplify{{dop} - { dop1} * t},$
we require

$\simplify{{dop} - { dop1} * t}=0, \qquad\qquad t=\var{sfrac(lort[0],lort[1])}.$

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This question is used in the following exam:

UA 1 - Unassessed Assignment 1.1 by Ian Davies in _MA-142.

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Vector Algebra - Practice-2.1

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Ian Davies 3 years, 3 months ago

Ian Davies 3 years, 3 months ago

Ian Davies 3 years, 3 months ago

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Vector Algebra - Practice-2.1

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Ian Davies 3 years, 3 months ago

Ian Davies 3 years, 3 months ago

Ian Davies 3 years, 3 months ago

Error in variable testing condition

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Warning

Generated value: string

Parts

Gap-fill

Generated value: `string`